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We present a fast algorithm for modular exponentiation when the factorizationof the modulus is known. Let $a,n,m$ be positive integers and suppose $m$factors canonically as $\prod_{i=1}^k p_i^{e_i}$. Choose integer parameters$t_i\in [0, e_i]$ for $1\le i\le k$. Then we can compute the modularexponentiation $a^n\pmod{m}$ in $O(\max(e_i/t_i)+\sum_{i=1}^k t_i\log p_i)$steps (i.e. modular operations). We go on to analyze this algorithmmathematically and programmatically, showing significant asymptotic improvementin specific cases.

An Elementary Method For Fast Modular Exponentiation With Factored Modulus